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Open research questions in Analytic and geometric function theory

97 gap statements mined from Analytic and geometric function theory papers in our 4.5M-paper local library, which holds 643 papers on the topic — drawn mostly from each paper's own stated research gap, future-work, challenge and limitation notes, and its abstract. The ones listed below are a selection still marked open; each names the study that raised it, with a DOI link where the paper has one.

What the literature leaves open

  • The paper suggests future research directions, such as the study of Riemannian surfaces with more general properties. The paper suggests the use of new methods to estimate the bi-Lipschitz constant, such as the use of other types of barriers.

    The bi-Lipschitz constant of an isothermal coordinate chart · 2026 · DOI

  • The paper identifies a gap in prior work, which is the lack of a quantitative estimate of the bi-Lipschitz constant. The paper identifies the need for a new method to estimate the bi-Lipschitz constant, which is based on the properties of harmonic barriers.

    The bi-Lipschitz constant of an isothermal coordinate chart · 2026 · DOI

  • The definition of strongly starlike functions is broad, making it difficult to find a general solution. The coefficients of strongly starlike functions can be complex, making it challenging to calculate them exactly. The lack of a single method to estimate the coefficients of all strongly starlike functions.

    On a certain subclass of strongly starlike functions · 2026 · DOI

  • The results are limited to the estimation of logarithmic coefficients. The paper does not provide a general solution for all strongly starlike functions.

    On a certain subclass of strongly starlike functions · 2026 · DOI

  • There is a need to extend Ozaki's condition. There is a need to determine new sufficient conditions for functions to be in the class of p-valent starlike of order α.

    On starlikeness of $p$-valent analytic functions · 2026

  • The paper suggests further study on the boundary Schwarz lemma for vector-valued holomorphic functions. The paper proposes investigating the rigidity theorem for holomorphic self-mappings of $B_{\ell_p^n}$.

    On the Boundary Schwarz lemma and the rigidity theorem for certain mappings · 2026

  • The lack of estimates for coefficient bounds of regular functions using a q-series approach. The need for a novel set of regular q-functions to study complex functions.

    Four-Leaf Inequalities: A q-Series Approach · 2026 · DOI

  • The lack of a complete description of the solutions to the anisotropic partitioning problem. The need for sharp isoperimetric inequalities with respect to convex cones. The need for a second variation formula for the anisotropic area of hypersurfaces with non-empty boundary.

    On the Anisotropic Partitioning Problem in Euclidean Convex Domains · 2026 · DOI

  • The lack of sharp coefficient bounds and Hankel determinants for the class S * cos. The need for a deeper understanding of starlike functions and their applications.

    Sharp inverse logarithmic coefficient bounds for starlike functions associated with cosine function · 2026 · DOI

  • The paper does not provide a comprehensive comparison with other existing methods. The results are limited to the subclass of analytic functions defined on a petal-shaped domain.

    Analysis of certain determinants for a defined subclass of analytic functions using Poisson distribution series in petal shaped domain · 2026 · DOI

  • The paper suggests further research into coefficient functional problems related to Poisson distribution series of analytic functions. The paper proposes the study of other subclasses of analytic functions defined on different domains.

    Analysis of certain determinants for a defined subclass of analytic functions using Poisson distribution series in petal shaped domain · 2026 · DOI

  • The study of coefficient problems within the novel subclass is identified as a research gap. The analysis of Hankel determinants for the subclass is also identified as a gap.

    Properties of some univalent functions associated with balloon-shaped domain · 2026 · DOI

  • The borderline case p = 1 was previously unsolved. There was a lack of sharp rotation bounds for homeomorphisms with integrable distortion and controlled compression.

    Pointwise rotation for homeomorphisms with integrable distortion and controlled compression · 2026 · DOI

  • Further study of the properties of the function φc(z, α) is suggested. Further study of the applications of the results to geometric functions is suggested. Further study of the extensions of the results to other classes of functions is suggested.

    Sharp Differential Subordinations for Cardioid Starlike Functions via Hypergeometric Techniques · 2026 · DOI

  • The paper identifies a need for the derivation of sufficient conditions for analytic functions to belong to a new Ma-Minda class of cardioid-starlike functions.

    Sharp Differential Subordinations for Cardioid Starlike Functions via Hypergeometric Techniques · 2026 · DOI

  • The authors note that the classical nodal geometry may not survive after passing from a single eigenfunction to a finite spectral sum.

    Nested nodal loops for sums of Laplace eigenfunctions · 2026

  • To investigate whether the results can be extended to other generalized growth scales and growth-oscillation frameworks. To explore the relationships between different growth theories and the oscillatory behavior of solutions of complex differential equations. To apply the results to other areas of complex analysis.

    Growth of (alpha, beta, gamma)-order solutions to complex linear differential equations with analytic coefficients in the unit disc · 2026 · DOI

  • To extend the results of the paper to higher dimensions. To compare the performance of the proposed method with other methods. To apply the results of the paper to other areas of mathematics and computer science.

    On the Banach-Mazur Ellipse · 2026 · DOI

  • The paper identifies a gap in the theory of functions in q-theory. The paper aims to fill this gap by analyzing the Hadamard product of two power series.

    Hadamard product of convex functions and Jackson operator · 2026

  • To study the Dirichlet problem for higher-order iterated Vekua equations. To develop new methods for solving the Dirichlet problems for the complex components of the idempotent representation of a bicomplex function. To study the properties of bicomplex functions and their applications.

    Generalizations of the Dirichlet problem for bianalytic functions · 2026

  • To study the topology of the nodal set of a linear combination of Laplace eigenfunctions in more general settings. To construct more general examples of biharmonic functions with nodal sets containing double nests. To understand the implications of the failure of the Boggio--Hadamard conjecture for the study of elliptic partial differential equations.

    Nested nodal loops for sums of Laplace eigenfunctions · 2026

  • To study the behavior of anisotropic materials using the results of the paper. To optimize the shape of objects with respect to a surface tension using the results of the paper. To study the behavior of biological systems with anisotropic properties using the results of the paper.

    On the Anisotropic Partitioning Problem in Euclidean Convex Domains · 2026 · DOI

  • The paper identifies a gap in the understanding of the oscillation of Fourier coefficients of automorphic forms. The paper aims to fill this gap by establishing a quantitative result regarding the sign changes of the sequence of Fourier coefficients.

    ON THE QUANTITATIVE SIMULTANEOUS SIGN CHANGES OF FOURIER COEFFICIENTS ATTACHED TO HECKE-MAASS FORMS OVER CERTAIN QUADRATIC FORMS · 2026 · DOI

  • The study identifies a gap in the existing literature regarding the growth of solutions to complex linear differential equations with analytic coefficients in the unit disc. The paper aims to fill this gap by establishing new results and generalizing earlier works.

    Growth of (alpha, beta, gamma)-order solutions to complex linear differential equations with analytic coefficients in the unit disc · 2026 · DOI

  • Exploring the classes involving higher order derivatives. Studying the properties of the defined function classes QM q (φ) and CC q (g, φ).

    Starlike functions associated with Q-multiplicative derivative · 2026 · DOI

Most-cited papers in Analytic and geometric function theory

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Open questions raised by papers listed on this page include: (1) The paper identifies a gap in prior work, which is the lack of a quantitative estimate of the bi-Lipschitz constant (“The bi-Lipschitz constant of an isothermal coordinate chart”, 2026); (2) The lack of a single method to estimate the coefficients of all strongly starlike functions (“On a certain subclass of strongly starlike functions”, 2026); (3) The lack of estimates for coefficient bounds of regular functions using a q-series approach (“Four-Leaf Inequalities: A q-Series Approach”, 2026); (4) The lack of a complete description of the solutions to the anisotropic partitioning problem (“On the Anisotropic Partitioning Problem in Euclidean Convex Domains”, 2026); (5) The lack of sharp coefficient bounds and Hankel determinants for the class S * cos (“Sharp inverse logarithmic coefficient bounds for starlike functions associated…”, 2026). 97 gap statements have been mined from Analytic and geometric function theory papers in our 4.5M-paper local library, which holds 643 papers on the topic; the gaps come from whichever of those papers state one. They are mostly the research gaps the authors state and the papers' abstracts, plus future-work, limitations and challenges passages. The ones listed below are a selection still marked open; each names the study that raised it, with a DOI link where the paper has one, so you can read the original claim in context.

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